Key Concepts and Formulas

• Bernoulli Differential Equation: A differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$, where $n$ is a real number not equal to 0 or 1.
• Substitution for Bernoulli Equation: To solve a Bernoulli equation, use the substitution $v = y^{1-n}$.
• Integrating Factor: For a linear first-order differential equation $\frac{dv}{dx} + P_1(x)v = Q_1(x)$, the integrating factor is given by $I.F. = e^{\int P_1(x) dx}$.

Step-by-Step Solution

Step 1: Identify and Transform the Bernoulli Equation

The given differential equation is $\cos x\,dy = y\left( {\sin x - y} \right)dx,\,\,0 < x <{\pi \over 2}$. We need to rewrite it in the standard Bernoulli form.

• Divide by $\cos x \, dx$: $\frac{dy}{dx} = \frac{y(\sin x - y)}{\cos x}$ This isolates $\frac{dy}{dx}$ on one side.
• Separate terms and rewrite using trigonometric identities: $\frac{dy}{dx} = y\frac{\sin x}{\cos x} - y^2\frac{1}{\cos x} = y\tan x - y^2\sec x$ We use $\frac{\sin x}{\cos x} = \tan x$ and $\frac{1}{\cos x} = \sec x$ to simplify.
• Rearrange to Bernoulli form: $\frac{dy}{dx} - (\tan x)y = -(\sec x)y^2$ This matches the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$, where $P(x) = -\tan x$, $Q(x) = -\sec x$, and $n = 2$.
• Divide by $y^2$: $\frac{1}{y^2}\frac{dy}{dx} - \frac{\tan x}{y} = -\sec x$ This prepares the equation for the substitution.
• Substitute $v = \frac{1}{y}$: Let $v = \frac{1}{y} = y^{-1}$. Then, $\frac{dv}{dx} = -\frac{1}{y^2}\frac{dy}{dx}$. So, $\frac{1}{y^2}\frac{dy}{dx} = -\frac{dv}{dx}$. This substitution will transform the equation into a linear one.
• Substitute into the equation: $-\frac{dv}{dx} - (\tan x)v = -\sec x$ We replace $\frac{1}{y^2}\frac{dy}{dx}$ with $-\frac{dv}{dx}$ and $\frac{1}{y}$ with $v$.
• Multiply by $-1$ to get the standard linear form: $\frac{dv}{dx} + (\tan x)v = \sec x$ Now we have a linear first-order differential equation in the form $\frac{dv}{dx} + P_1(x)v = Q_1(x)$, where $P_1(x) = \tan x$ and $Q_1(x) = \sec x$.

Step 2: Solve the Linear Differential Equation

• Find the Integrating Factor (I.F.): $I.F. = e^{\int P_1(x)dx} = e^{\int \tan x dx}$ The integrating factor helps us solve the linear equation.
• Evaluate the integral: $\int \tan x dx = \int \frac{\sin x}{\cos x} dx = -\ln|\cos x| = \ln|\sec x|$ Since $0 < x < \frac{\pi}{2}$, $\cos x > 0$ and $\sec x > 0$, so we can drop the absolute value.
• Calculate the I.F.: $I.F. = e^{\ln(\sec x)} = \sec x$
• Multiply the linear equation by the I.F.: $(\sec x)\frac{dv}{dx} + (\sec x)(\tan x)v = \sec^2 x$ This makes the left side a derivative of a product.
• Recognize the left side as a derivative: $\frac{d}{dx}(v\sec x) = \sec^2 x$
• Integrate both sides with respect to $x$: $\int \frac{d}{dx}(v\sec x) dx = \int \sec^2 x dx$ $v\sec x = \tan x + C$ This gives us the general solution in terms of $v$ and $x$.

Step 3: Substitute Back to the Original Variable

• Substitute $v = \frac{1}{y}$ back into the equation: $\frac{1}{y}\sec x = \tan x + C$ This expresses the solution in terms of the original variable $y$.
• Rearrange the equation: $\sec x = y(\tan x + C)$ $y = \frac{\sec x}{\tan x + C}$
• Rewrite the equation: $y = \frac{\sec x}{\tan x + C}$ $\sec x = (\tan x + C)y$

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs when rearranging and substituting in the Bernoulli equation. A simple sign error can throw off the entire solution.
• Integrating Factor: Remember to integrate $P_1(x)$ correctly to find the integrating factor. The integral of $\tan x$ is $\ln|\sec x|$. Don't forget the absolute value, but consider whether it can be dropped based on the given domain.
• Back Substitution: Always substitute back to the original variable y after solving for v.

Summary

We solved the given Bernoulli differential equation by first transforming it into a linear first-order differential equation using the substitution $v = \frac{1}{y}$. We then found the integrating factor, solved for $v$, and finally substituted back to express the solution in terms of $y$. The final solution is $\sec x = (\tan x + C)y$.

The final answer is \boxed{\sec x = \left( {\tan x + c} \right)y}, which corresponds to option (D).